[BioC] edgeR: interactions in linear models

[Ryan C. Thompson]

Dear Gordon,

I understand now. The 3-way experiment cannot be reduced to 2-way 
marginals, but one can still test for 2-way interactions within each 
level of the 3rd factor. That matches with my own previous 
understanding, and my confusion was just one of terminology. Thank you 
for the clarification!

-Ryan

On Sun May 12 18:29:20 2013, Gordon K Smyth wrote:
> Dear Ryan,
>
> The marginality principle is most easily understood in a 2-way
> factorial model.  Suppose you have a 2x2 factorial experiment with two
> genotypes and two treatments (active vs control).
>
> If a two-way interaction exists, then this means that the treatment
> effect is different for the genotypes.  It makes no sense to test for
> a "treatment effect" in this situation (even though mathematical
> models allow you to do so) because there is no consistent treatment
> effect without specifying the genotype.
>
> On the other hand, it always meaningful to test for a treatment effect
> in the two genotypes separately, and then to ask whether the two
> treatment effects are consistent or different.
>
> In a 3-way factorial model, a 3-way interaction means that the
> experiment cannot be reduced to 2-way marginals in any meaningful way.
>
> Best wishes
> Gordon
>
>
> On Mon, 13 May 2013, Gordon K Smyth wrote:
>
>> On Sun, 12 May 2013, Ryan C. Thompson wrote:
>>
>>> Hi Gordon,
>>>
>>> In a previous email on the list you said:
>>>
>>>> testing for the 2-way interaction in the presence of a 3-way
>>>> interaction does not make statistical sense.  This is because the
>>>> parametrization of the 2-way interaction as a subset of the 3-way
>>>> is somewhat arbitrary. Before you can test the 2-way interaction
>>>> species*treatment in a meaningful way you would need to accept that
>>>> the 3-way interaction is not necessary and remove it from the model.
>>>
>>> Does this mean that it is impossible to test for a 2-way interaction
>>> when your model includes a 3-way interaction term?
>>
>> It is mathematically possible but has no scientific meaning.  This is
>> called the marginality principle in linear models:
>>
>> http://en.wikipedia.org/wiki/Principle_of_marginality
>>
>>> Or does it just mean that the parametrization provided by
>>> "model.matrix(~1+factor1*factor2*factor3)" is such that the 2-way
>>> interaction is not represented by any coefficient, but rather by a
>>> complex contrast?
>>
>> The same principle applies regardless of the parametrization.
>>
>>>> I prefer to fit the saturated model (a different level for each
>>>> treatment combination) and make specific contrasts. There is some
>>>> discussion of this in the limma User's Guide.
>>
>>> If I understand correctly here, you are saying that one can fit a
>>> model where each coefficient represents the abundance for one
>>> specific combination of the 3 factors, as in the Limma User's Guide
>>> section 8.5.2 "Analysing as for a Single Factor". In other words,
>>> one could do "model.matrix(~0+factor1:factor2:factor3)" and this
>>> would be an alternate parametrization of the same design. And with
>>> this parametrization, all the 2- and 3-way interaction terms (and
>>> simple pairwise comparisons) can easily be tested from the single
>>> full 3-way interaction by specifying the appropriate contrasts. Do I
>>> understand correctly?
>>
>> Yes, I am recommending the group mean parametrization, as in the
>> limma User's Guide Section 8.5.2 or edgeR User's Guide Section 3.3.1.
>>
>> I recommend this parametrization because each contrast that is drawn
>> has an explicit meaning in terms of comparisons of groups and can be
>> interpretted on its own terms.
>>
>> The original poster did what I intended.
>>
>> Best
>> Gordon
>>
>>> -Ryan Thompson
>>>
>>
>
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